More precisely, how does one characterize integrally closed finitely generated domains (say, over C) based on geometric properties of their varieties? Given a finitely generated domain A and its integral closure A' (in its field of fractions), what's the geometric relationship between V(A) and V(A')?

Integral closure of the coordinate ring corresponds to normality of the space. Somebody told me that I shouldn't feel bad for not understanding the meaning of normality or normalization. (c.f. mathoverflow.net/questions/46/…)
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Anton GeraschenkoOct 20 '09 at 22:28

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"Normal" has got to be one of the most overloaded words in mathematics, and it's worse than usual here because it doesn't evoke any kind of geometric feeling.
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Qiaochu YuanOct 20 '09 at 22:35

2 Answers
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The property you are interested in is known as being normal. For affine varieties, the definition of normal is just that the coordinate ring is integrally closed, and the process of replacing a variety by its coordinate ring is known as normalization (a general variety is said to be normal if it is locally isomorphic to a normal affine variety.) So far I've just restated your question; but there are a number of things known. For this we need the notion of smoothness: for varieties over C, this should be equivalent to being a smooth manifold (the general definition is a bit technical).

1: Any smooth variety is normal.
2: The set of singular points of a normal variety has codimension >=2.
Corollary: For curves, normal <=> smooth.

Shafarevich's Basic Algebraic Geometry vol. 1 is a good reference for this from the varieties point of view (and deals with smoothness more rigorously).

As regards the relationship between the varieties corresponding to A and A': I mostly just have intuition for curves, so I'll stick to talking about them. For curves, A' is a version of A with the singularities "resolved": more specifically, V(A') is a smooth variety equipped with a surjective morphism of varieties from A'->A which is an isomorphism away from the preimages of the singular points of A. (This should be true in higher dimensions too I think: it's definitely true if one is talking about schemes, but I think it's also true that for affine varieties the map A->A' induces a map V(A')->V(A).) The two basic examples to keep in mind here are the cuspidal cubic C_1: y^2 = x^3 and the nodal cubic C_2 y^2 = x^3 + x^2.

In the case C_1: the coordinate ring C[x, y]/(y^2 - x^3) has integral closure isomorphic to C[t], and the map of varieties here is the map from the affine line to C_1 given by t|-> (t^3, t^2). In this case the map is a bijection as sets (but not an isomorphism of affine varieties! because the inverse map cannot be expressed as a polynomial map), and the "cusp" of C_1 that is visible at the point (0,0) is no longer evident.

In the case C_2: the coordinate ring also has integral closure isomorphic to C[t]: this time the map is a bit more complicated, but it's t|-> (t^2 -1 , t(t^2-1)). How did I find that? In this case, looking at the curve one sees that it has a self-intersection at the origin. This means that there should be two distinct points in the normalization that have been sent to the same point in C_2. Another way of stating that is that because the curve appears to have two tangent lines at the origin, there really should be two different points there, one on each tangent line. How to tell them apart? Well, as one approaches the origin from one direction, the ratio y/x tends to 1 in the limit, whereas if one approaches it from the other direction, the ratio y/x tends to -1: so at one of our two points, y/x=1, and at the other one, y/x = -1. Since y/x is well-defined everywhere else on the curve, this suggests that we want t = y/x to belong to our coordinate ring at the origin. Indeed, t^2 = x +1, so t is integral, and we can solve for x and y in terms of t to get the original answer. So in this case we have a surjective map from the affine line to a self-intersecting curve which is injective everywhere except at the preimage of the singular point at the origin.

Just to be clear, normal<=>smooth works for affine varieties over arbitrary field (not necessarily alg. closed), right? That is, coordinate rings of smooth affine curves are Dedekind domains regardless of the base field. Is that correct?
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PavelCMay 5 at 9:15

This is normalization. As for characterizing them, over C, in the complex topology on the variety, think of it as meaning (roughly, not exactly) locally irreducible. Most important is that normal varieties are smooth in codimension one, and so you have an easier time talking about divisors on them. And, a follow-up for others. In the complex topology, is normal equivalent to smooth in codim 1 and locally irreducible? These criteria handle all the cases I can come up with off the top of my head.

Alright. So am I right to say that if A is the ring of functions on a curve C, then A is integrally closed if and only if C is nonsingular? If it is singular, what does the variety of its integral closure look like?
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Qiaochu YuanOct 20 '09 at 22:28

@Qiaochu: the variety of the integral closure is exactly the desingularization. You can get it by blowing up any singular points if you want.
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Anton GeraschenkoOct 20 '09 at 22:32

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@Charles: If you take the affine cone over a rational quartic in PP^3, this is not normal, and I believe that this provides a counterexample to your question. More concretely, let I be the kernel of the map K[x_0,x_1,x_2,x_3]--> K[s,t] by sending the variables to s^4,s^3t,st^3 and t^4. Then K[x_0,..,x_3]/I is not normal (consider s^2t^2 from the quotient field), but the corresponding affine variety is irreducible and smooth in codim 1.
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Daniel ErmanOct 21 '09 at 6:42

But look locally in the complex topology. Doesn't it analytically factor at the singular point?
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Charles SiegelOct 21 '09 at 11:40